## ripple

Some time ago, I tried coming up with my own water-drop/ripple simulation for making a small animated-gif background (akin to the moving backgrounds, like clouds, ocean breakers, etc, that are so popular as church lyric-projector backgrounds right now). Basically, I had a sinusoidish curve, decaying with radius and time; I simulated a dozen or so of these drips, usually centered off-picture, and added the curves; it made for some nice randomish moving water-surface. (Sorry, I cannot find where I put it for now.)

Anyway, I did a search, and found a YouTube video for a ripple using
z=(cos( 0.5sqrt(x^2+y^2)-6n)/(0.5(x^2+y^2)+1+2n)), for n={0…10}. Based on a couple of quick
spreadsheet graphs, it appears that n is the time element.

{jsxgraph ripple}

Note that this z-function has a similar response curve to sinc(t) = sin(t)/t = s(t), but damps much more quickly. I think what I did in mine was base it on s(t)*e^(-kt).

{jsxgraph ripple vs sinc}

It looks like with the appropriate scaling on t, and choice of k, my since-with-exponential-decay could closely replicate the video’s curve. I guess mine wasn’t as poor of a choice as I thought at one time. Assuming that’s what I really did.